986.311
The maximum limit set of identical facets into which
any system can be
divided consists of 120 similar spherical right triangles
ACB whose three corners are 60
degrees at A, 90 degrees at C, and 36 degrees at B.
Sixty of these right spherical triangles
are positive (active), and 60 are negative (passive).
(See Sec.
901.)

986.312
These 120 right spherical surface triangles are described
by three different
central angles of 37.37736814 degrees for arc AB, 31.71747441
degrees for arc BC, and
20.90515745 degrees for arc AC__which three central-angle
arcs total exactly 90 degrees.
These 120 spherical right triangles are self-patterned
into producing 30 identical spherical
diamond groups bounded by the same central angles and
having corresponding flat-faceted
diamond groups consisting of four of the 120 angularly
identical (60 positive, 60 negative)
triangles. Their three surface corners are 90 degrees
at C, 31.71747441 degrees at B, and
58.2825256 degrees at A. (See Fig.
986.502.)

986.313
These diamonds, like all diamonds, are rhombic forms.
The 30-symmetrical-
diamond system is called the rhombic triacontahedron:
its 30 mid-diamond faces (right-
angle cross points) are approximately tangent to the
unit-vector-radius sphere when the
volume of the rhombic triacontahedron is exactly tetravolume-5.
(See Fig.
986.314.)

986.314
I therefore asked Robert Grip and Chris Kitrick to
prepare a graphic
comparison of the various radii and their respective
polyhedral profiles of all the
symmetric polyhedra of tetravolume 5 (or close to 5)
existing within the primitive cosmic
hierarchy (Sec.
982.62)
__i.e. other than those of tetravolumes
1, 2, 3, 4, and 6__which
carefully drafted drawing of the tetravolume-5 polyhedra
(and those polyhedra
"approximately" tetravolume S) my colleagues did prepare
(see Fig.
986.314). These
exactly tetravolume-5 polyhedra are, for example__

the icosahedron with outer edges of unit vector
length;

the icosahedron of outer vertex radius of unit vector
length;

the regular dodecahedron of unit vector edge; and

the regular dodecahedron of unit vector radius

__all of which show that they have only a slightly greater
radius length than that of the
prime vector.

986.315
The chart of the polyhedral profiles (Fig.
986.314)
shows the
triacontahedron of tetravolume 5 having its mid-diamond-face
point C at a distance
outward radially from the volumetric center that approximately
equals the relative length
of the prime vector. I say "approximately" because the
trigonometrically calculated value
is .999483332 instead of 1, a 0.0005166676 radial difference,
which__though possibly
caused in some very meager degree by the lack of absolute
resolvability of trigonometric
calculations themselves__is on careful mathematical review
so close to correct as to be
unalterable by any known conventional trigonometric
error allowance. It is also so correct
as to hold historical significance, as we shall soon
discover. Such a discrepancy is so
meager in relation, for instance, to planet Earth's
spheric diameter of approximately 8,000
miles that the spherical surface aberration would be
approximately the same as that
existing between sea level and the height of Mount Fuji,
which is only half the altitude of
Mount Everest. And even Mount Everest is invisible on
the Earth's profile when the Earth
is photographed from outer space. The mathematical detection
of such meager relative
proportioning differences has time and again proven
to be of inestimable value to science
in first detecting and then discovering cosmically profound
phenomena. In such a context
my "spherical energy content" of 4.99, instead of exactly
5, became a thought-provoking
difference to be importantly remembered.

986.316
By careful study of the Grip-Kitrick drawings of tetravolume-5
polyhedra it
is discovered that the graphically displayed zones of
radial proximity to one another of all
the tetravolume-5 symmetric polyhedra (Fig.
986.314)
describe such meager radial
differences at their respective systems' outermost points
as to suggest that their
circumferential zone enclosed between the most extremely
varied and the most inwardly
radiused of all their axially spun vertexes of the exact
tetravolume-5 polyhedra may
altogether be assumed to constitute the zone of limit
cases of radiantly swept-out and
pulsating tetravolume-5 kinetic systems.

986.317
Recognizing that polyhedra are closed systems and that
there are only seven
cases of symmetrical subdivisioning of systems by the
most economical great-circle
spinnings (and most economically by the chords of the
great-circle arcs), we discover and
prove structurally that the maximum-limit abundance
of a unit-symmetrical-polyhedral-
system's identical facetings is the rhombic triacontahedron,
each of whose 30 symmetrical
diamond planar faces may be symmetrically subdivided
into four identical right triangles
(30 × 4=120), and we find that the triacontahedron's
120-spherical-right-triangled frame
of system reference is the maximum-limit case of identical
faceting of any and all
symmetrical polyhedral systems in Universe. This maximum-limit-system
structuring proof
is accomplished by the physically permitted, great-circle-spun,
hemispherical self-halvings,
as permitted by any and all of the seven cosmic limit
cases of symmetric systems' being
spun-defined around all the respective system's geometrically
definitive (ergo, inherent)
axes of symmetrical spinnability. It is thus that we
learn experimentally how all the
symmetric systems of Universe self-fractionate their
initial system unities into the
maximum number of omniangularly identical surface triangles
outwardly defining their
respective internal-structure tetrahedra whose angles-central
or surface-are always
independent of a system's time-size considerations.
And because they are independent of
time-size considerations, such minimum-maximum limit-case
ranges embrace all the
symmetrical polyhedral systems' generalized-primitive-conceptuality
phenomena.

986.400 T Quanta Module

986.401 Consideration 12: Dynamic Spinning of Rhombic Triacontahedron

986.402
I then speculated that the only-by-spinning-produced,
only superficially
apparent "sphericity" could be roundly aspected by spinning
the rhombic triacontahedron
of tetravolume 5. This rational volumetric value of
exactly 5 tetravolumes placed the
rhombic triacontahedron neatly into membership in the
primitive hierarchy family of
symmetric polyhedra, filling the only remaining vacancy
in the holistic rational-number
hierarchy of primitive polyhedral volumes from 1 through
6, as presented in Table
1053.51A.

986.404
In the case of the rhombic dodecahedron the mid-diamond-face
point C is
exactly PVR distance from the polyhedral system's volumetric
(nucleic) center, while in
the case of the rhombic triacontahedron the point C
is at approximately PVR distance from
the system's volumetric (nucleic) center. The distance
outward to C from the nucleic
center of the rhombic dodecahedron is that same PVR
length as the prime unit vector of
the isotropic vector matrix. This aspect of the rhombic
triacontahedron is shown at Fig.
986.314.

986.405
The symmetric polyhedral centers of both the rhombic
dodecahedron and the
rhombic triacontahedron may be identified as 0, and
both of their respective external
diamond faces' short axes may be identified as A-A and
their respective long axes as B-B.
Both the rhombic dodecahedron's and the triacontahedron's
external diamond faces ABAB
and their respective volumetric centers O describe semiasymmetric
pentahedra
conventionally labeled as OABAB. The diamond surface
faces ABA of both OABAB
pentahedra are external to their respective rhombic-hedra
symmetrical systems, while their
triangular sides OAB (four each) are internal to their
respective rhombichedra systems.
The angles describing the short A-A axis and the long
B-B axis, as well as the surface and
central angles of the rhombic dodecahedron's OABAB pentahedron,
all differ from those
of the triacontahedron's OABAB pentahedron.